The solution to the strong CP problem of this \(Z_{\cal N}\)  scenario required \({\cal N}\)  to be odd.
Overall, the \(\sim 10\)  orders of magnitude tuning required by the SM strong CP problem is traded by a \(1/{\cal N}\)  adjustment, where \({\cal N}\)  could be as low as \({\cal N}=3\)  (viable solutions were found with \(3\le {\cal N}\lesssim 47\)  for \(f_a\lesssim 2.4\times 10^{15}\)  GeV, and any \({\cal N}\)  for larger
values of \(f_a\)  [1]}).
The resulting axion is exponentially lighter than the canonical QCD one, because the non-perturbative contributions to its potential from the \({\cal N}\)  degenerate QCD groups conspire by symmetry to suppress each other.
This can be intuitively understood from the large \({\cal N}\)  limit of a non-linearly realized
\(Z_{\cal N}\)  shift symmetry, which is a continuous global \(U(1)\)  symmetry: the
axion acts as the Goldstone boson of the discrete symmetry [2]}. Its mass thus vanishes asymptotically for large \({\cal N}\)  as befits a \(U(1)\) 
Goldstone boson. Indeed, it has been shown [1]} that in the large \({\cal N}\)  limit the total axion potential is given in all generality by
\(V_{\mathcal {N}}\left(\theta _a\right)\simeq - \frac{m_a^2 f_a^2}{{\cal N}^2} \,\cos ({\cal N}\theta _a)\,,\) 
